Previous works [1]}, [2]}, [3]} have studied the error threshold of the RHG cluster state model when qubits are erased with some probability.
This is a natural noise model in optics and bears some resemblance to our model, as one assumes that the locations of the erasures are known.
The relationship between the erasure threshold and the \(Z\)  error threshold was found to be approximately linear [1]} and there is a fundamental erasure threshold of \(0.249\) , which is set by percolation theory [5]}.
It is difficult to directly compare our results with those of [1]} because of the differences in the noise models.
However, our swap-out threshold is close to the percolation theory erasure threshold, and it is natural to ask whether we can increase the swap-out threshold beyond the erasure threshold by further optimizing our decoder.
There are many ways we could improve our current decoder (see Sec:Open), so, unless there is a fundamental limit due to percolation of swap-outs, we are hopeful that we can surpass the erasure threshold.
In addition, the question as to whether our decoder has an advantage over the equivalent erasure decoder for finite values of \(\Delta _{\rm dB}\)  remains open and could be a subject of future work.
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